L(s) = 1 | + i·2-s + 4-s − 3i·7-s + 3i·8-s − 11-s − 2i·13-s + 3·14-s − 16-s + 3i·17-s + 19-s − i·22-s − i·23-s + 2·26-s − 3i·28-s + 6·29-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.5·4-s − 1.13i·7-s + 1.06i·8-s − 0.301·11-s − 0.554i·13-s + 0.801·14-s − 0.250·16-s + 0.727i·17-s + 0.229·19-s − 0.213i·22-s − 0.208i·23-s + 0.392·26-s − 0.566i·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.202883293\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.202883293\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - iT - 2T^{2} \) |
| 7 | \( 1 + 3iT - 7T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - iT - 37T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 - 11T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 5T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 17iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.568891059817690710786278515823, −8.148868382659839341113182116194, −7.34048424876465677186142570469, −6.79806317192970297973470887521, −6.02171445368533873754317562258, −5.20833064497245747081884276596, −4.29102506892960570404081863746, −3.28092545924390481765675384592, −2.24582127773753597897398860312, −0.893115655868234040583118614749,
1.04255413523686930824297306454, 2.34941860132668614952611307400, 2.73066940417275293644769737710, 3.84998293964607259201682175877, 4.89858419998148600373127449043, 5.77506754366716819444239097594, 6.55760243283327021991045613731, 7.28023622010360050458234235152, 8.220659555474796358667056744293, 9.017041722458801716994923028904